Eccoci.

Given an $n$-tuple of numbers another $n$-tuple is created where each element of the new $n$-tuple is chosen randomly from the numbers in the previous $n$-tuple. For example, given $(2,2,3)$ the probability that $2$ occurs in the first position in the next 3-tuple is $2/3$. The probability of getting all $2$'s would be $8/27$ while the probability of getting the same 3-tuple (in any order) would be $4/9$.

Let $E(n)$ be the expected number of steps starting with $(1,2,\ldots,n)$ and ending with all numbers being the same.

You are given $E(3) = 27/7$ and $E(5) = 468125/60701 \approx 7.711982$ rounded to 6 digits after the decimal place.

Find $E(10^3)$. Give the answer rounded to 6 digits after the decimal place.